Can this also apply to the dollar bill…I am trying to do something like that with 1$.

It’s pretty easy if you think about it, and there isn’t a whole load of maths involved.
Start simple: 1, 2, 3, 4, 5, 6 = 1 + 2 + 3 + 4 + 5 + 6
Now rearrange them: (1+ 6) + (2 + 5) + (3 + 4)
Which is obviously 7 + 7 + 7
Which is also 3 * 7
Since you always start with one, you will always get N / 2 “pairs” (the center “pair” will be just a single digit if it’s an odd number of terms): 6 / 2 = 3
And you need to include the total of each pair: 6 + 1 = 7
so N / 2 pairs, of N +1
Which gives you N * (N + 1) / 2
Or 6 * (6 + 1) / 2 = 6 * 7 / 2 = 42 / 2 = 21
So for 356, it’s exactly the same process, just with bigger numbers: 365 * (365 + 1) / 2 = 365 * 366 / 2 = 133,590 / 2 = 66,795 pennies.
(I used a calculator for this bit!)

Alternatively you could just:

ceil(365 / 2) * 365 = 66.795

However, this doesn’t work for even numbers :( I wrote this small Python program that can handle both even and odd numbers and compares against the result from the accumulation approach:

x = 0
z = 365
# z = 14

# By accumulation
for y in range(1, z + 1):
x += y

print(‘Accumulated result : %s’ % x)

# By calculation
if z % 2 == 0:
# Even numbers
print(‘Calculated result(Even): %s’ % (math.floor(z / 2) * (z + 1)))
else:
# Odd numbers
print(‘Calculated result(Odd) : %s’ % (math.ceil(z / 2) * z))

This is how I arrive at the answer. Lets do 1-10 first. Take the last number in sequence e.g. number 10. Divide this by 2 = 5. Multiply this by the last number = 50 and then add the 5. So the sequence will add up to 55. So to recap 10/2 x 10 + 5 = 55.
Now for 1-365 adding 1 everyday of the year. So last number is 365/2 = 182.5
So 365×182.5 + 182.5 = £66,795

Hah! Yeah – need those parenthesis around that first part huh :)

The only issue I have with this is the image at the top, that formula is written incorrectly. Surprised no other math whizzes picked up on it!

Not sure what you mean exactly, but def. encourage you to try the 52 week challenge :) Good thing is that you can start it anytime you want! And the money adds up fast, as you can see from some of the examples here.

You can find it here :

http://www.wikihow.com/Sum-the-Integers-from-1-to-N